Exercise 3.1
page no 3.26
Type 1
Question 1
Examine which of the following relations are functions? If they are functions, find their domain and range :
(i) $f=\{(1,4),(2,3),(3,4),(4,3)\}$
(ii) $g=\{(2,5),(-1,0),(1,6)\}$
(iii) $h=\{(1,2),(2,2),(3,2)\}$
(iv) $\phi=\{(1,2),(1,3),(2,5)\}$
(v) $\psi=\{(2,1),(3,1),(5,2)\}$
(vi) $u=\{(2,1),(5,1),(8,1),(11,1),(14,1),(17,1)\}$
(vii) $y=\{(0,0),(1,1),(1,-1),(4,2),(4,-2),(9,3),(9,-3),(16,4),(16,-4)\}$
Question 2
Let $A=\{1,2,3\}$ and $B=\{4,5\}$.
Let $f=\{(2,4),(3,5)\}$. Is $f$ a function from $A$ into $B$ (from $A$ to $B$ ).
Question 3
Let $A=\{1,2,3\}, B=\{3,6,9,10\}$. Which of the following relations are functions from $A$ to $B$ ? Also find their range if they are functions.
$f=\{(1,9),(2,3),(3,10)\}$
$g=\{(1,6),(2,10),(3,9),(1,3)\}$
$h=\{(2,6),(3,9)\}$
$u=\{(x, y): y=3 x, x \in A\}$
Page no 3.27
Question 4
Let $A=\{1,2,3\}, B=\{4,5\}$
Let $f=\{(1,4),(1,5),(2,4),(3,5)\}$. Is $f$ a function from $A$ to $B$ ?
Question 5
Let $A=\{a, b, c, d\}$. Examine which of the following relations is a function on $A$ ?
(i) $f=\{(a, a),(b, c),(c, d),(d, c)\}$
(ii) $g=\{(a, c),(b, d),(b, c)\}$
(iii) $h=\{(b, c),(d, a),(a, a)\}$
Question 6
Let $A=\{1,2,3,4\}, B=\{1,5,9,11,15,16\}$
and $f=\{(1,5),(2,9),(3,1),(4,5),(2,11)\}$
Is $f$ a function from $A$ to $B$.
Question 7
(i) Let $f$ be the subset of $Q \times Z$ defined by :
$f=\left\{\left(\frac{m}{n}, m\right): m \in Z, n \in Z, n \neq 0\right\}$. Is $f$ a function from $Q$ to $Z$ ? Justify your answer.
(ii) The relation $f$ is defined by $f(x)= \begin{cases}x^{2}, & 0 \leq x \leq 3 \\ 3 x, & 3 \leq x \leq 10\end{cases}$
The relation $g$ is defined by $g(x)= \begin{cases}x^{2}, & 0 \leq x \leq 2 \\ 3 x, & 2 \leq x \leq 10\end{cases}$
Show that $f$ is a function and $g$ is not a function.
[Hint : $g(2)=2^{2}=4$ and $g(2)=3 \times 2=6 \quad \therefore 2$ has two images 4 and 6 ]
Question 8
Let $N$ be the set of natural numbers and the relation $R$ be defined on $N$ such that $R=\{(x, y): y=2 x, x, y \in N\}$. Find the domain, codomain and range of $R$. Is this relation a function.
Type 2
Question 9
(i) A function $f$ is defined by $f(x)=2 x-5$, find $f(0), f(7)$ and $f(-3)$
(ii) If $f(x)=x^{2}+4, x \in R$ then, find $f(2), f(-5)$ and $f(t+2), t \in R$
(iii) If $f(x)=x^{2}+1, x \in R$, find $f(1) \times f(4)$
Question 10
If $f: R \rightarrow R$ be defined by $f(x)=x^{2}+2 x+1$. Then find :
(i) $f(-1) \times f(1)$ Is $f(-1)+f(1)=$ $f(0) ?$ (ii) $f(2) \times f(3)$. Is $f(2) \times f(3)=f(6) ?$
Question 11
(i) Let $f=\{(1,1),(2,3),(0,-1),(-1,-3)\}$ be a function from $Z$ to $Z$ defined by $f(x)=a x+b$ for some integers $a, b$, determine $a$ and $b$.
(ii) Let $f=\{(1,1),(2,3),(0,-1),(-1,-3)\}$ be a linear function from $Z$ to $Z$, find $f(x)$.
Question 12
Which of the following relations are functions ? Give reasons. If it is a function, determine its domain and range.
(i) $f=\{(2,1),(3,1),(4,2)\}$
(ii) $g=\{(1,3),(1,5),(2,5)\}$
(iii) $h=\{(2,2),(2,4),(3,3),(4,4)\}$
(iv) $\varphi=\{(2,1),(5,1),(8,1),(11,1),(14,1),(17,1)\}$
Type 3
Question 13
Function $f$ is given by $f=\{(4,2),(9,1),(6,1),(10,3)]$. Find the domain and range of $f$.
Question 14
Let $A={9,10,11,12,13}$ and let $f: A \rightarrow N$ be a function defined by $f(n)=$ the highest prime factor of $n$. Find the range of $f$.
Question 15
If $A=\{-3,-2,-1,0,1,2,3\}$ and $f^{\prime}(x)=x^{2}-1$ defines $f: A \rightarrow R$. Then find range $f$
Question 16
Find the domain and range of the following functions.
(i) $f(x)=x$
(ii) $f(x)=(x-1)$
(iii) $f(x)=x^{2}-1$
(iv) $f(x)=x^{2}+2$
(v) $f^{\prime}(x)=\sqrt{x-1}$
Question 17
Find the domain of the function $f(x)=\frac{x^{2}+3 x+5}{x^{2}-5 x+4}$
Question 18
Find the range of the function $f(x)=2-3 x, x>0$
Question 19
Find the domain and range of the following functions:
(i) $\left\{\left(x, \sqrt{9-x^{2}}\right): x \in R\right\}$
(ii) $\{(x,-|x|): x \in R\}$.
Question 20
Find the domain of definition and range of the function defined by the rules:
(i) $f(x)=x^{2}$
(ii) $g(x)=|x|$
(iii) $h(x)=\frac{1}{3-x^{2}}$
(iv) $u(x)=\sqrt{4-x^{2}}$
Question 21
Consider the following rules :
(i) $f: R \rightarrow R: f(x)=\log _{c} x$
(ii) $g: R \rightarrow R: g(x)=\sqrt{x}$
(iii) $h: A \rightarrow R: h(x)=\frac{1}{x^{2}-4}$, where $A=R-\{-2,2\}$
Which of them are functions? Also find their range, if they are functions.
Question 22
Let $f: R-\{2\} \rightarrow R$ be defined by $f(x)=\frac{x^{2}-4}{x-2}$ and $g: R \rightarrow R$ be defined by $g(x)=x+2$ Find whether $f=g$ or not.
Type 4
Question 23
Let $f: R \rightarrow R$ and $g: R \rightarrow R$ be defined by $f(x)=x+1, g(x)=2 x-3$.
Find $f+g, f-g$ and $\frac{f}{g}$
Question 24
Let $f=\{(1,1),(2,3),(0,-1),(-1,-3)\}$ be a linear function from $Z$ into $Z$ and $g(x)=x$. Find $f+g$.
Question 25
Find $f+g, f-g, f \cdot g, \frac{f}{g}$ and $\alpha f(\alpha \in R)$ if
(i) $f(x)=\frac{1}{x+4}, x \neq-4$ and $g(x)=(x+4)^{3}$
(ii) $f(x)=\cos x, g(x)=e^{x}$
(iii) $f(x)=\sqrt{x-1}, g(x)=\sqrt{x+1}$
Question 26
If $f(x)=x, g(x)=|x|$, find $(f+g)(-2),(f-g)(2),(f \cdot g)(2)$,
$\left(\frac{f}{g}\right)(-2), 5 f(2)$
Type 5
Question 27
Draw the graph of the following functions
(i) $f: R \rightarrow R$ such that $f(x)=x-1$
(ii) $f: R \rightarrow R$ such that $f(x)=|x-1|$
(iii) $f: R \rightarrow R$ such that $f(x)=|x-2|$
(iv) $f: R \rightarrow R$ such that $f(x)=4-2 x$
Question 28
Let $N$ be the set of all natural numbers.
Define a real valued.
function $f: N \rightarrow N$ by $f(x)=2 x+1$.
Using this definition complete the table given below :
$\begin{array}{|cccccccc|}\hline x & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\hline f(x) & f(1)= & f(2)= & f(3)= & f(4)= & f(5)= & f(6)= & f(7)= \\\hline\end{array}$
Question 29
Define the function $f: R \rightarrow R$ by $y=f(x)=x^{2}$. Complete the table given below :
(Image to be added)
Find the domain and range of function $f$ and draw its graph.
Question 30
Define the real valued function on $f: R-\{0\} \rightarrow R$ as $f(x)=\frac{1}{x}$ Complete the figure given below:
(Image to be added)
Find the domain and range of $f$.
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